time-series clustering for data analysis in smart grid · and power constraints. one of the popular...

Time-Series Clustering for Data Analysis in Smart Grid

Akanksha Maurya* Alper Sinan Akyurek* Baris Aksanli** Tajana Simunic Rosing**

*Electrical and Computer Engineering Department, University of California San Diego (UCSD)

**Computer Science and Engineering Department, University of California San Diego (UCSD)

Abstract—The increasingly pervasive deployment of

networked sensors in the Smart Grid for monitoring energy

consumption has resulted in an unprecedentedly large amount of

data generation. Efficient methods are required to understand

this high volume and high dimensional data on an embedded

platform, which has many challenges due to memory, processing

and power constraints. One of the popular methods to analyze

time-series data is clustering. In this paper, we discuss D-Stream

II, a common time-series clustering algorithm, and demonstrate

that it fails to obtain clusters in sample Smart Grid applications.

Then, we propose an enhanced version of this algorithm, which

handles the scenarios where the original algorithm fails. We show

the effectiveness of our algorithm using real residential power

consumption data from Pecan Street database. Our enhanced

algorithm efficiently handles the high volume energy

consumption data and captures the everyday energy

consumption patterns of each residential home, which allow the

consumer to compare energy consumption with their neighbors

or detect any abnormality such as a defective appliance. The

consumers can also be clustered into different groups, which can

be effectively used to enhance the demand response policies. Our

algorithm can perform clustering on approximately half a million

energy consumption data points using only 5KB max memory,

360J max overhead in energy consumption and can complete in 8

mins on a resource limited embedded platform (Raspberry Pi 2).

Keywords— time-series, clustering, smart grids, density-based

clustering, embedded platform

I. INTRODUCTION AND RELATED WORK

Smart Grid leverages a large number of sensors to collect

information about the surroundings, e.g. energy, voltage,

current of each appliance, temperature data, etc. This generates

a huge volume of data and the need for faster processing [1].

Such continuously increasing data provide tremendous

opportunities in understanding the dynamics of the consumer

as well as the utility to optimize the Smart Grid. Data can be

generated at a high temporal resolution (e.g. every second), but

frequently, relatively low-resolution data (e.g., every 15 mins)

is transmitted and used, as the amount of storage space required

for high-resolution data is prohibitively large. For utility

companies serving millions of customers, storing this high-

resolution data can sum up to petabytes [12]. Similarly,

transmission of such high-resolution data can cause serious

congestion issues in the transmission network. However, high-

resolution data is extremely beneficial for many analytical

applications such as detailed visualization [13], energy

disaggregation [14], monitoring of residential power quality

[15], and short-term forecasts of energy generation/load [16,

17] or energy prices [18]. Effective techniques are required for

analyzing this data closer to the source (e.g., a smart meter) so

that relatively low volume of data is stored and transmitted. In

the Smart Grid the lower-end devices closer to where data is

generated are typically embedded devices, which has

constraints in memory, processing speed and power.

Time-series clustering on sensor data can be highly

beneficial for Smart Grid applications, such as analyzing

everyday energy demand patterns. Both users and utilities can

take advantage of the results of such analysis. The former

group can optimize their energy consumption by adjusting their

demand based on electricity price and/or alternative energy

resource availability. The latter can identify potential user

groups by clustering, which can be effectively used to enhance

the Demand Response policies [3] for a real-time automatic

control in the Smart Grid.

Time-series clustering methods are classified into five

categories [4]: Partitioning Clustering, Hierarchical Clustering,

Density-based Clustering, Grid-based Clustering, and Model-

based Clustering. An effective clustering algorithm for an

embedded platform is one which performs clustering

accurately, using limited input from the user with the memory,

processing and power constraints of the embedded system.

CluStream [5] is a partitioning based clustering algorithm for

time-series data. It uses k-means as the base method and forms

spherical shaped clusters. This algorithm fails to form clusters

of arbitrary shape and also requires the user to pre-set the

number of clusters. Thus, it is not appropriate for our

application. CluTree [6] is a hierarchical clustering based

algorithm for time-series data. This algorithm has no

backtracking capability, i.e. once a cluster is merged or split, it

cannot be undone. Similar to the previous one, it has very

limited adaptability. In hierarchical clustering, cluster

information is stored for every hierarchy level, which makes

CluTree very expensive in terms of memory utilization.

SWEM [7] is a model-based clustering algorithm that clusters

data in a time-based sliding window with expectation

maximization technique. This algorithm optimizes the fit

between the data and a mathematical model. The accuracy of

the algorithm depends heavily on the accuracy of the pre-

selected mathematical model. The algorithms that are based on

partitioning, hierarchical clustering and model-based clustering

are not appropriate for embedded platforms due to their non-

adaptive behavior [4].

DenStream [8] is a density-based clustering algorithm. It

has the ability to form arbitrary shape clusters, detect outliers

and automatically determine the number of clusters. Often

2016 IEEE International Conference on Smart Grid Communications (SmartGridComm): Data Management and Grid Analyticsand Dynamic Pricing

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density-based clustering is combined with grid-based

clustering, which is known as density-grid-based clustering [4].

D-Stream [9] is a density-grid-based time-series clustering

algorithm. This algorithm has fast processing time and

bounded memory utilization since it depends on the number of

grids, which is fixed, instead of the number of data points. D-

Stream II [10] is an extension of the original D-Stream

algorithm. It improves the clustering accuracy by considering

the attraction of grids, which characterizes the spatial

information of the data in each grid. Due to low memory

utilization, fast processing time and clustering accuracy [4][10]

as compared to other algorithms, D-Stream II is an appropriate

fit for clustering time-series data on an embedded platform

such as Raspberry Pi 2 (RPi2) in a Smart Grid. RPi2 provides a

low-cost platform which interconnects and controls various

devices/sensors and presents a computing environment that

suits well for embedded Smart Grid applications.

Despite its advantages, there are three scenarios which

occur frequently in real time series, where the D-Stream II

algorithm fails or is inaccurate. For example, D-Stream II

algorithm fails to generate any clusters for 135 real time-series

datasets of one-year energy consumption of 15 residential

house from Pecan Street, which results in significant

information loss and leads to an incorrect interpretation of

energy consumption data. In this paper, we propose a modified

D-Stream II algorithm, which performs clustering in scenarios

where the original algorithm fails while maintaining the

performance in other scenarios. Our algorithm performs

clustering on approximately half a million energy consumption

data points using only 5KB max memory, 360J max energy

consumption overhead and can complete in 8 minutes on RPi2

which is within the memory, processing and power constraints

of RPi2.

The remainder of this paper is organized as follows: in the

next section, we introduce the original D-Stream II algorithm

and its issues. In section III, we introduce our clustering

algorithm. In section IV, we analyze one-year energy

consumption data of 15 residential houses from Pecan Street

with our algorithm. Finally, in section V, we conclude the

time-series clustering for analysis in Smart Grid.

II. D-STREAM II ALGORITHM

The D-Stream II algorithm has online and offline phases. In

the online phase as shown in Algorithm 1, lines 5-7, each

multi-dimensional input data is mapped to a corresponding

discretized grid and the characteristic vector of the grid is

updated. In the offline phase, which is performed after every

time interval gap as shown in Algorithm 1, lines 8-14, the

clusters are adjusted dynamically i.e. new clusters are created

and existing clusters are disintegrated. The D-Stream II

algorithm adopts a density decaying technique to capture the

dynamic changes of a data stream and an attraction-based

mechanism to accurately generate cluster boundaries. It

clusters two neighboring grids only if they are strongly

correlated. Two grids are considered as strongly correlated if

their attractions in both directions are higher than a threshold

value. During the offline phase, it only adjusts grids whose

density attributes changed since the last time the grids were

adjusted.

Each grid stores a characteristic vector which captures the

evolving nature of the data stream. The characteristic vector of

a grid g is a tuple (tg, tm, C, D, label) [17], where tg is the last

time when g is updated, tm is the last time when g is removed

from grid list as a sporadic grid, C is a 2D-vector denoting the

attraction from g to its neighbors, D is the grid density at the

last update, and label is the class label of the grid. Whenever a

data point is mapped to the grid its characteristic vector is

updated.

Algorithm 1: Main Procedure of D-Stream II

1. tc = 0;

2. initialize an empty red-black tree for grid list;

3. while data stream is active do

4. read record x = (x1, x2, . . . , xd);

5. determine the density grid g that contains x;

6. if (g not in grid list) insert g to grid list;

7. update the characteristic vector of g;

8. if tc == gap then

9. call initial clustering(grid list);

10. end if

11. if tc mod gap == 0 then

12. detect and remove sporadic grids from grid list;

13. call adjust clustering(grid list);

14. end if

15. tc = tc + 1;

16. end while

The algorithm takes into consideration the attraction of two

grids while merging the grids into a cluster. As shown in Fig 1,

the data in grid 1 is located at the upper left corner and the data

in grid 5 is located at the lower half of the grid and both the

grids have density above the threshold value, while the density

of grids 2, 3, 4 and 6 is less than dense threshold. The other

algorithm will cluster grid 1 and grid 5 and consider other grids

as transitional or sparse. The D-Stream II algorithm will cluster

grid 1 with grid 2, 3, 4 and grid 5 with grid 6 based on the

attraction of the neighboring grids which based on the spatial

locality of the data looks more accurate.

Fig 1 Attraction between different grids[17]

Algorithm 2 : Procedure for initial clustering in D-Stream II (grid list)

1. update the density of all grids in grid list;

2. assign each dense grid to a distinct cluster;

3. label all other grids as NO CLASS;

4. repeat

5. foreach cluster c

6. foreach outside grid g of c

7. foreach neighboring grid h of g

8. if (g and h are strongly correlated) and (h

belongs to cluster c’)

9. if (|c| > |c’|) label all grids in c’ as in c;

10. else label all grids in c as in c’;

11. else if (g and h are strongly correlated) and

(h is transitional) label h as in c;

12. until no change in the cluster labels can be made


Algorithm 2 shows the first component of offline phase i.e.

initial clustering which is called when the time interval is equal

to the gap time. The initial clustering procedure marks all the

dense grids as separate clusters and for each cluster checks if a

strongly correlated neighbor grid exists or not. If a strongly

correlated neighbor is found then the neighboring grid is

merged with the cluster.

Algorithm 3: Procedure for adjust clustering in D-Stream II (grid list)


2. foreach grid g whose attribute

(dense/sparse/transitional) is changed since last

call to adjust clustering()

3. if (g is a sparse grid)

4. delete g from its cluster c, label g as NO CLASS;

5. if (c becomes unconnected) split c into two clusters;

6. else if (g is a dense grid)

7. among all neighboring grids of g that are strongly

correlated to g, find out the grid h whose cluster

ch has the largest size;

8. if (h is a dense grid)

9. if (g is labeled as NO CLASS) label g as in ch;

10. else if (g is in cluster c and |c| > |ch|)

11. label all grids in ch as in c;

12. else if (g is in cluster c and |c| ≤ |ch|)

13. label all grids in c as in ch;

14. else if (h is a transitional grid)

15. if ((g is NO CLASS) and (h is an outside grid

if g is added to ch)) label g as in ch;

16. else if (g is in cluster c and |c| ≥ |ch|)

17. move h from cluster ch to c;

18. else if (g is a transitional grid)

19. among neighboring grids of g that are dense and

strongly correlated to g, find the grid whose

cluster c’ has the largest size;

20. label g as in c’;

21. end for

The second component of offline phase i.e. adjusts

clustering procedure is shown in Algorithm 3. Adjust

clustering procedure is called whenever the time interval is a

multiple of the gap time. The algorithm clusters two

neighboring grids only if they are strongly correlated where

two grids are considered as strongly correlated if their

attractions in both directions are higher than a threshold value.

The threshold value is calculated by averaging the total sum of

attraction between each pair of grids which is 1

|𝑃|(1−𝜆) , where P

is the total number of grid pairs and λ is decaying factor.

During the offline phase, the algorithm only adjusts grids

whose density attributes changed since the last time the grids

were adjusted. For example, at time t1 grid g is transitional and

after tgap time, the grid becomes dense then the algorithm

adjust the grids and clusters the grid g with its strongly

correlated neighbor. But, if even after tgap time the grid g is

still transitional, the algorithm does not adjust the grids.

The D-Stream II algorithm determines, this time, interval

gap so that the dynamic nature of the data is captured and the

time interval is not too small or large. The D-Stream II

algorithm handles outliers by detecting the sporadic grids from

the sparse grid and removing the grid from the grid list. The

sparse grid is considered as sporadic grid if the density of the

grid is less than the density threshold function which is

determined such that a transitional or dense grid will never be

falsely deleted due to the removal of the sporadic grid. This

approach improves the time complexity of the algorithm since

all grids are not adjusted every time gap interval.

Although the original D-Stream II algorithm performs

accurate and efficient time-series data clustering in most

scenarios, it fails or is inaccurate in three scenarios: 1) no

isolated dense clusters are formed after initial clustering as the

original D-Stream II algorithm generates a newly isolated

clusters only once in offline phase i.e. only in the initial

clustering procedure, 2) dense grids are never merged if

initially they are not correlated as the original D-Stream II

algorithm does not consider correlation as an attribute and 3)

the original D-Stream II algorithm also considers the diagonal

grids along with the neighboring grids for calculation attraction

with neighboring grids.

III. MODIFIED D-STREAM ALGORITHM

Our modification of D-Stream II clustering algorithm

handles all three scenarios mentioned above while maintaining

the performance in other scenarios and performs effective and

accurate clustering as compared to the original D-Stream II

algorithm. It generates a new isolated dense grid in every

offline phase call, it also considers the correlation between the

two grids as clustering criteria and only considers the

neighboring grids for attraction calculation. The algorithm is

also capable of detecting and removing sporadic grids mapped

by outliers in order to dramatically improve the space and time

efficiency of the system.

Algorithm 4 : Main procedure for our modified algorithm

1. tc = 0;

2. initialize an empty red-black tree for grid list;

3. while data stream is active do

4. read record x = (x1, x2, . . . , xd);

5. determine the density grid g that contains x;

6. if (g not in grid list) insert g to grid list;

7. update the characteristic vector of g;

8. if (tc mod gap == 0) or (tc == gap )then

9. detect and remove sporadic grids from grid list;

10. call adjust clustering(grid list);

11. end if

12. tc = tc + 1;

13. end while

A. Isolated dense grids:

The original D-Stream II algorithm considers an isolated

dense grid as a newly generated cluster only in the initial

clustering procedure. During the subsequent call of offline

phase, D-Stream II algorithm checks if a dense grid has any

strongly correlated neighboring grids which belong to an

already generated cluster or not. If such a strongly correlated

neighbor is found then the grid is added to the neighboring

cluster. If no such neighboring grid is found, then the D-Stream

II algorithm does not categorize the dense grid into a newly

generated cluster which over the time can grow into a larger

cluster. Since D-Stream II algorithm does not consider such

isolated dense grids the clustering result are not very accurate.

In our modified D-Stream II algorithm, we consider an isolated

dense grid as a newly generated cluster if no strongly

correlated neighbors are found. Algorithms 4 and 5 show the


main and adjust clustering procedures of the modified D-

Stream II algorithm. In the modified D-Stream II algorithm,

adjust clustering procedure for every time interval gap, if for a

dense grid g a strongly correlated neighbor does not exist and

the grid does not already belong to any cluster then a newly

isolated cluster is formed as shown in Algorithm 5 in line 9 and

10. Since in the modified algorithm we always generate an

isolated cluster for a dense grid in adjust clustering procedure,

we do not have to explicitly call the initial clustering

procedure. Therefore, in our modified D-Stream II clustering

algorithm the call for the initial clustering procedure is

removed from the main procedure as shown in Algorithm 4 on

line 8-10.

Algorithm 5: Adjust clustering procedure in our algorithm (grid list)


2. foreach grid g whose attribute

(dense/sparse/transitional) is changed since last

call to adjust clustering() or the grid becomes

strongly correlated to at least one neighbor

3. if (g is a sparse grid)

4. delete g from its cluster c, label g as NO CLASS;

5. if (c becomes unconnected) split c into two clusters;

6. else if (g is a dense grid)

7. among all neighboring grids of g that are strongly

correlated to g, find out the grid h whose cluster

ch has the largest size;

8. if (h is not found) and (g does not belong to any

cluster)

9. assign grid g to a distinct cluster;

10. else if (h is a dense grid)

11. if (g is labeled as NO CLASS) label g as in ch;

12. else if (g is in cluster c and |c| > |ch|)

13. label all grids in ch as in c;

14. else if (g is in cluster c and |c| ≤ |ch|)

15. label all grids in c as in ch;

16. else if (h is a transitional grid)

17. if ((g is NO CLASS) and (h is an outside grid

if g is added to ch)) label g as in ch;

18. else if (g is in cluster c and |c| ≥ |ch|)

19. move h from cluster ch to c;

20. else if (g is a transitional grid)

21. among neighboring grids of g that are dense and

strongly correlated to g, find the grid whose

cluster c’ has the largest size;

22. label g as in c’;

23. end for

B. Dense grid merging attributes:

D-Stream II algorithm adjusts the grid list only when the

grid attributes change since the last time the grid was adjusted.

The algorithm only considers the whether a grid is dense,

transitional or sparse as an attribute. It does not take into

consideration the evolving attraction of the two grids. It is

possible that at time t two grids are dense but are not strongly

correlated as shown in Fig 2(a). However, over time, the two

grids become strongly correlated. The D-Stream II algorithm

only adjusts cluster when an attribute changes, it does not

merge the two dense grids as the attribute of both the grids has

not changed as shown in Fig 2(b). Thus, in our modified D-

Stream II algorithm, we also consider the correlation between

the grids as an attribute as shown in line 2 of algorithm 5 to

improve the clustering accuracy as shown in Fig 2(c).

(a)Non-correlated grids (b) D-Stream II (c) Our algorithm

Fig 2 Correlation between two grids

C. Attraction Calculation:

In D-Stream II algorithm the attraction of a grid with its

neighboring grids is calculated as shown in Fig 3(a) where the

attraction of grid g with grid f is defined as the ratio of the

volume of intersection of the hypercube (ACEG) yellow region

and grid f i.e FG*GH to the volume of the hypercube.

However, the D-Stream II algorithm defines the neighboring

grids of grid g as a set of grids whose center differ from g in at

most one direction. Therefore, a grid g will have 2 neighbors in

ith dimension, one whose center in ith dimension is greater than

the center of grid g in ith dimension and other with center in ith

dimension less than the center of grid g in ith dimension and the

center of grid g and its neighbor is equal in all another

dimension except ith dimension. In our modified D-Stream II

algorithm, while calculating the attraction of grid g to its

neighboring grids, we do not consider the attraction of grid g

with grid k as only grid f and grid h are neighbors of grid g as

shown in Fig 3(b) which improves the clustering accuracy.

(a) (b)

Fig 3 Grid attraction calculation

IV. RESULTS

We use our modified D-Stream II time-series clustering to

analyze everyday energy demand patterns. We analyze the real

time-series energy consumption data of 15 different residential

houses from Pecan Street which represent various people,

demographics, and energy consumption patterns. We used

three different time resolutions: 1 hour, 15 minutes and 1

minute and three different energy consumption resolutions:

0.125kWh, 0.25kWh, and 0.5kWh, where energy resolution is

equal to the width of each grid. Higher time and energy

resolution give more granular information about the energy

consumption pattern but utilize more memory. Lower time and

energy resolution give coarse information about the energy

consumption patterns but utilize less memory. We use different

time and energy resolution to analyze the trend between

granularity and memory utilization and observed that our D-

Stream II algorithm perform clustering within the memory

constraints of time and energy resolution as 1min and

0.125kWh respectively.

We implemented the modified D-Stream II clustering

algorithm on RPi2 which is a low processing embedded

platform and is very frequently used for home automation in a

Smart Grid environment. We analyzed a total of 135 different


data sets of real energy consumption 9 datasets per house for

different time and energy resolutions. The original D-Stream II

algorithm was not able to converge and provide clusters on any

of 135 data sets. However, our modified D-Stream II algorithm

obtained clusters for all 135 data sets of 15 different residential

house.

A. Comparison with original D-Stream II:

Fig 4 shows the number of grids used and the number of

clusters formed over time for clustering one-year energy

consumption data of house number 1 using original D-Stream

II and modified D-Stream II clustering algorithm for 1-hour

time resolution and 0.25kWh energy resolution. The number of

grids populated is the same for both the algorithm as it

represents the mapping of data which is same for both

algorithms. Fig 4 shows that the number of the clusters formed

is always zero for original D-Stream II but the modified D-

Stream II algorithm generates dynamic clusters over time

where the number of clusters first increases as the grid is being

populated for the first time and then decreases as correlated

clusters are merged and less frequently used ones are deleted.

Fig 4 Number of grids used and clusters formed over time

B. Applications:

The dynamic clustering can be used for context analysis in

many ways. One way is to analysis the frequency of different

energy consumption values over time. Fig 5 shows the number

of times a grid with particular energy consumption value is

updated over a time period of one-year for five representative

houses for the time resolution of 15 minutes and energy

resolution of 0.125kWh. Most frequent energy consumption for

house no 1, 2, 3, 4 and 5 is around 0.625kWh, 0.375kWh,

0.375kWh, 0.5kWh, and 0.25kWh respectively. The range of

most frequent energy consumption for house no 59, 68, 86, 93

and 94 is around 0.375-2.25 kWh, 0.25-1 kWh, 0.25-1.75 kWh,

0.25-1.5 kWh and 0.125-0.75 kWh respectively.

Fig 5 Frequency of energy consumption

Consumers can detect defective appliances by using

dynamic clustering. If different clusters with higher energy

values are formed for similar energy consumption pattern

which this means an appliance is consuming more energy

because the appliance is defective. They can also compare their

energy consumption pattern with their neighbor by comparing

the clusters formed by different houses. When customers

received information on the energy consumption of their

neighbors, average energy use declined [19].

C. Memory utilization on RPi2:

Fig 6 Max memory in bytes for different time resolutions

In Fig 6, we show the maximum memory usage in bytes

while clustering energy consumption data of each house for

different time and energy resolution. We can see that the

maximum memory utilization among all the houses is 4196

bytes (< 5KB) which occur for house no. 6 for energy

resolution of 0.125kWh and time resolution of 15 mins. Also,

we can see that the memory utilization increases with increase

in the time resolution as we will have more data points since

the sampling rate is increased and decreases with a decrease in

energy resolution as we will have less number of grids.

D. Power/Energy Consumption on RPi2:

a) Time resolution of 1 min

b) Time resolution of 15 mins

c) Time resolution of 1 hour

Fig 7. Power consumption on Raspberry Pi 2


Fig 7 shows the power consumption for different time

resolution while performing clustering using our algorithm for

house no 7 which has the maximum memory utilization among

15 houses. The power consumption when RPi2 is idle is

800mW and it increases to 1500mW when our clustering

algorithm is performed. Therefore, the energy consumption

overhead increase (shown as the shaded area in Fig 7) on RPi2

for clustering using our algorithm for house no 6 is 360J, 20J,

and 4J for time resolution 1min, 15mins and 1 hour

respectively. Average energy consumption overhead increase

on RPi2 for 15 houses is 291.6J, 16.84J and 3.78J for time

resolution 1min, 15mins and 1 hour respectively. The energy

consumption increases with increase in the granularity of time

resolution as the clustering is performed on a larger set of data

points and takes more execution time. The max energy

consumption overhead increase (360J) in RPi 2 due to our

algorithm is reasonably low for a real-time embedded

application.

E. Performance on RPi2:

We also found that for the given one-year energy

consumption of five different house the maximum number of

grids used were 75 out of 200 and the maximum number of

clusters formed were 26. Fig 8 shows the execution time in

logarithmic scale for three different houses for different time

resolutions. The total execution time for clustering one-year

energy consumption data for house no 6 using our algorithm on

RPi2 is 520sec, 25sec, and 5sec for the time resolution of 1

min, 15 mins and 1 hour respectively. The average execution

time on RPi2 for all 15 houses is 421sec, 21sec, and 4.5sec for

the time resolution of 1 min, 15 mins and 1 hour. The

execution time increased with increase in the granularity of

time resolution as we had more data points as shown in Fig 8.

Execution time for the time resolution of 1 min is more than

execution time for the time resolution of 15 mins and 1 hour

(60mins). The max execution time (520sec) on RPi2 for

processing nearly half a million data points is within the

processing constraint of a real-time embedded system.

Fig 8 Execution time vs time resolution

V. CONCLUSION

In this paper, we proposed a modified D-Stream II density

grid-based clustering algorithm for context analysis of time-

series data in Smart Grid. We show the D-Stream II fails to

cluster in three different scenarios while our algorithm

converges and perform clustering. Based on our analysis of

energy consumption of 15 residential houses, we showed that

modified D-Stream II clustering algorithm performs the

clustering within memory and processing constraints of RPi2.

The modified D-Stream II algorithm perform clustering on

approximately half a million energy consumption data values

within a memory utilization of 5KB and execution time of

8min and energy consumption overhead of 360J.

ACKNOWLEDGEMENT

This work was supported in part by TerraSwarm, one of six

centers of STARnet, a Semiconductor Research Corporation

program sponsored by MARCO and DARPA. This work was

funded also by ARPA-E NODES project.

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